OPTIMAL HEDGING OF STOCK
PORTFOLIOS AGAINST FOREIGN
EXCHANGE RISK: THEORY AND
APPLICATIONS
Warren Bailey
Edward Ng
Rene M. Stulz
This paper applies a formula for the optimal hedge in a mean-variance framework to an investment in the Nikkei 225. It is shown that through hedging U.S.
investors can construct a portfolio long in the Nikkei whose dollar excess return
has the same volatility as the yen excess return of the Nikkei. There seems to be
little gain from improving estimates of the exposure of the Nikkei to the dollaryen exchange rate; in contrast, the performance of portfolios can be enhanced
substantially by obtaining better forecasts of exchange rate changes.
I. INTRODUCTION
The extent to which internationally
diversified portfolios should be hedged
against exchange rate risk has recently attracted a lot of attention among practitioners and academics.1 For hedging to be meaningful,
the foreign currency
exposure of the domestic return of a portfolio of common stocks has to be
predictable and nontrivial. If hedging is feasible, investors have to compare the
loss in expected return from hedging with the corresponding
decrease in the
volatility of the portfolio. In a mean-variance
framework, the gain or loss in
expected return from hedging has the same weight in the investor’s objective
function as does any change in expected return resulting from changes in portfolio weights. There is no difference between deciding how much to hedge and
deciding how much of any asset to add to a portfolio. Consequently,
in this
framework, the hedging decision is straightforward once the exposures and the
cost of hedging have been computed.
If the currency exposure of stocks in their own currency is trivial and hedging
Warren Bailey Assistant Professor of Finance, Cornell University; Edward Ng Lecturer, National
University of Singapore; Rene M. Stulz Ricklis Chair in Business, The Ohio State University.
Global Finance Journal,
ISSN: 1044-0283
3(2), 97-114
Copyright D 1992 by JAI Press, Inc.
All rights of reproduction in any form reserved.
98
GLOBAL FINANCE JOURNAL
3(2), 1992
has no cost, the optimal hedge ratio is -1. The hedge ratio is defined as the
dollar value of the position in foreign currency per dollar invested long in foreign stock. Hence, a hedge ratio of -1 implies that for each dollar invested
abroad, a dollar’s worth of foreign currency is borrowed or sold forward. The
evidence on the own currency exposure of foreign stocks is mixed for studies
that use weekly or monthly data. For instance, some studies find that yen
returns on the Nikkei 225 are exposed to the dollar/yen exchange rate, whereas
others argue that this exposure is trivial.2
The cost of hedging a portfolio against unanticipated exchange rate changes is
simply the cost of taking currency forward positions. Abstracting from transaction costs, the expected gain from a short forward yen position is the difference
between the forward price and the expected value of the yen at maturity of the
forward contract. For the first ten years following the introduction of floating
exchange rates, the consensus among economists was that the expected return
to a short forward position is either equal to zero or cannot be predicted to be
different from zero with any confidence. This consensus no longer exists, but it
seems to be largely responsible for the common wisdom that the optimal hedge
ratio should be computed without regard for the cost of hedging. The major
dissenting view is that of Black [5], who uses a theoretical asset pricing model to
show that, in equilibrium, hedging is costly and hence the optimal hedge ratio
must be smaller than one in absolute value.
In this paper, we use daily data on stock returns to evaluate the exposure of
the Nikkei 225 to the dollar/yen exchange rate. We show that, although this
exposure can be predicted using time-series models, it seems difficult to achieve
more variance reduction through forecasted hedge ratios than by simply assuming that the optimal hedge ratio is - 1. Hence, we report no convincing evidence
to suggest that a hedge ratio of -1 is not the minimum-variance
hedge ratio.
Using a hedge ratio of -1, it is possible to make an investment in the Nikkei
insensitive to unanticipated
changes in the dollar/yen exchange rate. Consequently, an American investor can have the same return volatility on an investment in the Nikkei as a Japanese investor can have. If hedging is costly, eliminating all currency exposure in a portfolio is not optimal. Based on historical
dollar/yen hedging costs, we show that taking into account hedging costs leads
to an optimal hedge ratio that differs substantially from -1.
The paper proceeds as follows: In the first section, we review the main determinants of the optimal hedge in a mean-variance
framework. In following
section, we estimate the dollar exposure of the Nikkei 225 using a variety of
methods. We then formulate in another section the optimal hedge for a meanvariance investor taking into account the hedging cost and show, based on
historical returns, the extent to which the optimal hedge ratio differs from -1.
Concluding remarks are presented in the last section.
The Theoretical
Determinants
of the Hedge Ratio.
In this section, we show that the optimal hedge satisfies a simple formula in a
mean-variance framework. This formula makes it possible to interpret and rec-
99
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
oncile the various arguments related to hedging the currency exposure of portfolios of stocks. We consider a world of N countries. In country j, there are Mi
risky assets. In each country, there is also a nominally riskless asset. Rj is the
instantaneous
return of the nominally riskless asset in country j. Let Iii be the
price in the currency of country j of the i-th asset in that country and ej the price
of the currency of country j in units of the currency of country 1. All prices,
including the exchange rates, and all Rj are assumed to be continuous functions
of S state variables, each following an Ito process. Rates of change of prices are
assumed to be conditionally lognormal. Hence, over the next instant, the distribution of returns is characterized by the vector of mean returns and by the
variance-covariance
matrix of returns.
With our notation, the instantaneous return of the nominally risk-free asset of
country j in terms of the currency of country 1 is
R. +
I
d”i
ej
Hence, if one takes a long position in the i-th risky asset of country j, a short
position in the nominally risk-free asset of that country, and a long position in
the nominally risk-free asset of country 1, the return on that portfolio3 is
-
R,dt
+ R,dt
The portfolio return is equal to the currency 1 return of an investment in asset i
financed abroad plus an investment in the risk-free asset of country 1. The
investment in asset i financed abroad has no cost in currency 1. The portfolio
return is not a function of unanticipated changes in the exchange rate since the
covariance between the rate of change of the exchange rate and the return of
asset i is known ex ante.4 In this sense, the portfolio corresponds to an investment in the i-th asset of country j instantaneously hedged against currency risk.
In constructing this portfolio, the hedge ratio is - 1 because each long position in
asset i is accompanied by a sale of a corresponding
unit of currency j of the
nominally risk-free asset of country j. A hedge ratio of -1 is not the minimumvariance hedge ratio if local currency returns, i.e., dZjilZji, are correlated with the
exchange rate since, by taking an additional hedge position to offset this
covariance, it is possible to obtain a portfolio with a lower variance. Let dB$IB$
be the return given in equation (2). We call the portfolio that yields that return a
zero net foreign investment portfolio since it implies that all foreign investment
is financed abroad. Note however that the same return could be achieved
through an investment in the foreign asset i and a short position in an instantaneous forward contract in currency j. This is because, if covered interest rate
parity holds, Rj - RI is equal to minus the forward premium and Xi + $-
R,
is equal to the return of a long position in currency j sold forward for curr!mcy 1.
100
GLOBAL FINANCE JOURNAL
3(2), 1992
If dB$‘B$ is the currency h return of a zero net investment portfolio long in
asset i of country j, ~~~/~~ and ~g~/~~ have a correlation coefficient of one. To see
that this is true, consider the covariance between the currency 1 return on a zero
net foreign investment portfolio long in asset i of country j and the currency h
return of a zero net foreign investment in asseti of country j. From the construction of a zero net investment portfolio long in asset i of country j, the only
stochastic variable in the return of such a portfolio is the currency j return of
asset i irrespective of the numeraire used to construct the portfolio. Hence, the
instantaneous
variance and covariances of zero net investment portfolios with
other zero net investment portfolios do not depend on the numeraire used to
construct them.5
It immediately follows from this discussion that two investors in different
countries who invest in identical portfolios of nominally risky assets using zero
net foreign investment portfolios have portfolios with identical instantaneous
return volatilities. The expected excess returns on these portfolios differ only to
the extent that the own-currency returns of nominally risky assets are correlated
with the rate of change of the exchange rate between the two countries. Hence,
if own-c~ren~
returns of nominally risky assets are not correlated with the rate
of change of exchange rates, mean-variance efficient frontiers of zero net foreign
investment portfolios are the same irrespective of the numeraire chosen to evaluate portfolios, If, further, there is no reward in terms of expected return to
currency speculation, then there is no reason to hold foreign nominally riskless
assets beyond the positions required to establish zero net foreign investment
portfolios. Holding only nominally risky assets plus the domestic risk-free asset
is optimal. Hence, if it is optimal not to take a position in foreign nominally
riskless assets in excess of the one required to finance investment in foreign
nominally risky assets all investors hold the same portfolio of nominally risky
assets irrespective of their country of residence. Consequently, mean-variance
investors located in different countries will have different portfolios of risky
assets only to the extent that they find it optimal to take positions in foreign
nominally risk-free assets beyond the positions required to finance investments
in foreign nominally risky assets. One can show more generally that portfolios of
risky assets differ only in their proportional holdings of no~nally
riskless assets, see Stulz [15].
We now consider the implications of our discussion for currency hedging in
equilibrium. So far, we assumed that investors choose portfolios in the meanvariance space of nominal returns. One would expect investors to care about real
rather than nominal returns; but if inflation is not volatile, the distinction between real and nominal returns is unimportant for portfolio choice. Suppose
that inflation has zero volatility in country 1, that consumption baskets are
similar across countries and the law of one price holds. In this case, country 1
returns are real returns for all investors in the world. Hence, we can choose the
currency of country 1 as the numeraire currency for all investors. If investors do
not differ in risk-tolerance
across countries, they all select the same portfolio
using the numeraire currency, Clearly, this means that in equilibrium they must
hold the world market portfolio. If nominally riskless assets do not belong to
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
101
that portfolio, no investor holds such assets. Hence, in equilibrium, investors
cannot borrow to hedge and no hedging takes place.
Alternative assumptions about consumption baskets lead to different results
about equilibrium hedging. To see this, suppose that instead of having all investors consume the same basket of goods, all investors of a given country consume
the same basket of goods but investors consume different baskets of goods
across countries.6 In this case, if inflation is deterministic in each country, a
country’s nominally risk-free asset is riskless in real terms for investors of that
country. Hence, if investors in a country are sufficiently risk averse, they would
like to invest some of their wealth in the nominally risk-free asset of their
country. To the extent that investors in one country want to be long in their
country’s nominally risk-free asset, it becomes possible for other investors to be
short in that asset and hence to hedge currency risk. In this case, hedging takes
place in equilibrium.
By making assumptions about what investors consume across countries and
about the stochastic process followed by the price of their consumption basket in
their local currency, one can derive a hedge ratio that holds in equilibrium.
Depending on the assumptions,
this hedge ratio may or may not be the same
across investors of different countries. Typically, however, if risk-tolerances,
consumption baskets and inflation processes vary across countries, there will be
no universal hedge ratio. If there is no universal hedge ratio, one can still derive
a hedge formula for investors in a particular country that does not depend on
expected returns. However, the usefulness of such a formula is limited since it
requires precise knowledge of the consumption
baskets, risk-tolerances
and
stochastic processes of inflation across countries. In this case, an investor specific hedge formula that relies only on expected returns and the variancecovariance matrix of returns is easier to implement since investors typically use
expected returns and the variance-covariance
matrix of returns as inputs in
portfolio allocation anyway.
We now investigate the optimal hedge ratio for an investor who uses currency
1 as the numeraire.7 Let WB be the fraction of wealth an investor from country I
chooses to invest in the i-th asset of country j using a zero net foreign investment
portfolio. We choose asset 0 to be the nominally risk-free asset. If the i-th asset is
asset 0, we define w:a as the fraction of wealth invested in a long position in that
asset. Straightforward manipulations show that the investor holds the following
portfolio8
-w = b - w~v-l&
V-$
where w is the (
column vector of investment
proportions in assets risky in currency 1. Assets risky in currency 1 are all nominally
risky assets and the assets nominally riskless in currencies other than currency 1.
a is the investor’s target expected return. p is a vector of expected excess returns
in currency 1. For the i-th asset in currency j, the expected excess return is
defined to be the expected excess return of a zero net foreign investment portCgjMj+(N-l))
GLOBAL FINANCE JOURNAL
102
3(Z), 1992
folio with a long investment in asset j over the risk-free return in currency 1. The
expected excess return for an investment in the nominal risk-free asset in currency i is Rj + pei - RI, where )L~. is the expected rate of change of the price of
currency j in terms of currency i. V is the variance-covariance
matrix of returns
of investments in zero net foreign investment portfolios of foreign risky assets
and of investments in foreign nominal risk-free assets.
Consider now the exposure of the optimal portfolio with respect to currency j,
We use the usual definition of exposure, namely the conditional covariance of
the portfolio’s return with the rate of change of ej divided by the conditional
variance of the rate of change of ej. Thus, the exposure has the interpretation of a
conditional regression coefficient in a regression of the portfolio’s return on the
rate of change of the exchange rate. This exposure can be computed as follows.
Let & be a vector corresponding
to the covariances of the rate of change of ej
with the returns on the zero net investment portfolios and the foreign nominally
riskless assets. Hence, I& is a column of matrix V. The exposure of the optimal
portfolio with respect to currency j is
qq-2i
=
= u,-:_vlei(a- R,)(pV-$)V-lp
ug2i(a- R1)(FVslF)(Rj+
/J++ -
Xl)
(4)
where uzj is the instantaneous
vola~~
of the rate of change of the price of
currency j. Let zuj be the investor’s portfolio excluding holdings of the domestic
risk-free asset and of the nominally riskless asset of country j in excess of the
amount of that asset borrowed to finance investment in country j. Equation (4)
can be rewritten after noticing that the left-hand side is the covariance of the
return of portfolio wi with the rate of change of the price of currency j, which we
define as oejwj. Let wjObe the holdings of the nominally risk-free asset of country
j in excess of the amount required to finance investment in country j. We can
rewrite equation (4) as
where p$ is the beta of the portfolio zo with respect to the price of currency j.
Note that the portfolio we used to derive the formula is one that holds positions
in zero net foreign investment portfolios. Hence, the fraction of investment in
the risk-free asset of country j given in equation (5) corresponds to the investment in that asset in excess of borrowing an amount equal to the initial investment in nominally risky assets of that country. For instance, if the fraction of
wealth invested in country j’s risky assets is 0.10, the investor takes a position in
the risk-free asset of that country equal to Wjo - 0.10 since he has to borrow 0.10
times his wealth in the nominally riskless asset of country j to finance his investment in that country when constructing a zero net investment portfolio.
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
103
The formula given in equation (5) states that the foreign currency exposure of
the portfolio is set equal to zero unless a forward contract on currency j has an
expected payoff different from zero. If the expected payoff of the forward contract is positive, the investor will choose to take a long position in the foreign
currency; if it is negative, he will choose to take a short position. Hence, the
optimal hedge ratio is greater than one in absolute value if a long position in a
forward contract has a negative expected payoff and it is smaller than one if it
has a positive expected payoff.
The optimal hedge formula has a simple interpretation. If the expected payoff
of a forward contract is zero, bearing foreign exchange risk amounts to adding
volatility to the portfolio without earning a reward in the form of a higher
expected return. Hence, in this case, the optimal portfolio has no foreign exchange exposure. If there is a payoff to bearing foreign exchange risk, the optimal portfolio has a position in foreign exchange that depends on the reward
from bearing such risk relative to the additional risk resulting from this position.
If the expected payoff from a long forward position is positive, less hedging
takes place since hedging is costly.
Equation (5) can be rewritten so that the exposure one has to compute is the
exposure of a portfolio with known distribution and weights that sum to one, as
opposed to the portfolio ~j of equation (5) whose weights are investor specific.
To rewrite equation (5), one needs to construct a portfolio with weights proportional to ~j that sum to one and then find out how much an investor with target
rate of return a would invest in that portfolio. Let pP denote the mean excess
return of such a portfolio and assume that the foreign investments of the portfolio are all financed abroad. In this case, equation (5) becomes
(5’)
where kf is equal to Rj + pej - R,. In equation (5’), the only quantity that requires
investor-specific
information is a, the target rate of return of the portfolio. All
other variables in equation (5’) can be obtained from the distribution of asset
returns. Equation (5’) yields the optimal currency j position for a mean-variance
efficient portfolio with expected excess return (Y - R, irrespective of the asset
universe from which the efficient portfolio is chosen. In the remainder of the
paper, we take portfolio P to be the Nikkei 225 and assume that the investors can
invest in P, dollar and yen risk-free assets. An alternative interpretation of our
results is that the Nikkei 225 is a component of portfolio P and that the other
components of that portfolio are uncorrelated with the dollar/yen exchange rate.
The Exposure
of the Nikkei 225.
To optimally hedge a portfolio of common stocks, one has to estimate its
exposure. Generally, the relevant portfolio is the tangency portfolio without the
investment in the foreign currency. Here, however, we restrict our attention to
the Nikkei 225 to examine the exposure of that portfolio to the dollar/yen exchange rate. There is no loss of generality if the yen rate of return of the Nikkei
104
GLOBAL FINANCE JOURNAL
3(2), 1992
225 is uncorrelated with the rate of change of the yen price of other currencies
besides the dollar.9 Hence, we have to compute the conditional beta of the dollar
return of the Nikkei 225 and then use that beta to hedge. We focus on the
conditional beta of the dollar return of an investment in the Nikkei financed in
dollars to facilitate comparison with other work. This conditional beta equals 1 +
p’(f), where v(f) is the exposure of the Nikkei 225 computed according to the
formula defined in the previous section, namely the exposure of a portfolio
invested in the Nikkei 225 financed by borrowing in yen.
In our analysis, we use a database of Nikkei 225 daily returns from the beginning of 1982 to the end of 1989. Saturday returns are added to Friday returns
when applicable. The return on the index is computed as log(&lI,_,) where It is
the index level on day f. We assume that hedging involves borrowing at the
three-month euro-yen rate. Consequently, for a day, the interest rate cost is the
three-month euro-yen rate divided by 360. Similarly, excess returns are defined
as excess returns over the three-month euro-dollar rate. Hence, the excess return
on an unhedged position in the Nikkei is
Y1t = b#t4
=
- l%(L,et-1)
(log& - loglt_l)
-
X$,t-1
+ (loge, -
loge,_,)
- Xs+i
where et is the dollar price of the yen at date f and Xs+i is the three-month eurodollar rate at date f - 1 divided by 360. To compute the optimal hedge at date f,
we would like to know the ratio
where the covariance and variance for the period from date f to f + 1 are computed
with all information available at date f. We first use as our conditional forecast of
&+i the in-sample estimate for &, where a period corresponds to one month and
returns are observed daily. To estimate the in-sample &, we would like to observe
exchange rates at the same points in time when we observe the market index.
Unfortunately,
this cannot be done with available published data sources. Instead, we estimate the in-sample pt using either the yen/dollar exchange rate
observed in Tokyo in the middle of the trading day or the yen/dollar exchange
rate observed in New York at three p.m. Neither source for exchange rates yields
exchange rate changes that are perfectly synchronous with the stock returns.
The results are reported in Table 1. For each year in the sample, we provide the
mean and standard deviation of the monthly exposures. It is immediately obvious that the standard deviation is large, suggesting that the estimated betas
change much over time. However, the average betas reported in Table 1 are not
significantly different from one. Whereas we do not report the monthly betas, it
is noteworthy that some of them exceed two or are negative and differ significantly from one.
105
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
Table 1
EXPOSURE ESTIMATES
Year
1
2
3
4
5
6
1982
1.2984
(0.2809)
1.2851
(0.3261)
1.3835
(0.5343)
0.9879
(0.6244)
1.1021
(0.4714)
0.6778
(0.6425)
0.8403
(0.6244)
1.2180
(0.2791)
1.0970
(0.5040)
1.1233
(0.2955)
1.1796
(0.2899)
1.1948
(0.4918)
0.7903
(0.5847)
1.0671
(0.4654)
0.6533
(0.8065)
0.7020
(0.5847)
0.9086
(0.2568)
0.9494
(0.5281)
1.4242
(0.4021)
1.2692
(0.3526)
1.3236
(0.8430)
0.9037
(0.4117)
1.1103
(0.7177)
1.2692
(0.8127)
0.8053
(0.4323)
1.1661
(0.2039)
1.1463
(0.6004)
1.4255
(0.4897)
0.9729
(0.3648)
1.4446
(0.8550)
0.7927
(0.4356)
1.0414
(0.6073)
0.9729
(0.7392)
0.8205
(0.4575)
1.1757
(0.3069)
1.1003
(0.6115)
1.3001
(0.2784)
0.8686
(0.2187)
1.3931
(0.3719)
1.0494
(0.1926)
1.0599
(0.4002)
0.8686
(0.4534)
0.8497
(0.3631)
1.2185
(0.1685)
1.1209
(0.3720)
1.1526
(0.2393)
0.7938
(0.2093)
1.1542
(0.3596)
0.8158
(0.3397)
0.9734
(0.3907)
0.7938
(0.4718)
0.7751
(0.3208)
0.9453
(0.2264)
0.9637
(0.3642)
1983
1984
1985
1986
1987
1988
1989
All
This table presents the yearly averages of estimates of the exposure of the dollar
return of the Nikkei to the rate of change of the dollar price of the yen. The
estimates are obtained using all daily returns within a month. Columns 1 and 2
report estimates using the dollar price of a three-month euro-yen deposit and
the dollar price of the yen reported respectively in Tokyo and in New York.
Columns 3 and 4 are similar to columns 1 and 2, except that the estimates are
Dimson-Fowler betas. Columns 5 and 6 report estimates using the dollar price of
the yen reported, respectively, in Tokyo and in New York. The standard deviations reported in parentheses are computed using each year’s observations. The
last two rows report averages across years of the variables.
Interestingly, exposures measured using exchange rates observed in Tokyo are
higher than are exposures measured using exchange rates observed in New
York.10 The explanation for this result is most likely that the exchange rate
returns observed in Tokyo are more contemporaneous
with the index returns
than are the exchange rate returns observed in New York. A lack of synchronization of index and exchange rate returns is equivalent to measuring the exchange
rate returns imprecisely and biases the regression coefficient downward.li
One
way to reduce the problem caused by a lack of synchronization is to estimate the
regression coefficients by including leads and lags of the exchange rate return.12
The third and fourth columns in Table 1 provide estimates of the regression
106
GLOBAL FINANCE JOURNAL
3(2), 1992
coefficient obtained by regressing the index return on the lagged, the contemporaneous, and the leading exchange rate return. In this case, the reported
coefficient is obtained by adding up the three estimated regression coefficients.
It turns out that if this procedure is used, the estimated exposures are much less
sensitive to where the exchange rate returns are observed.
In our procedure for estimating the relevant exposures, we assume that hedging is undertaken by borrowing at the three-month euro-yen offering rate and
repaying the amount borrowed on the next day. Hence, the exposures we are
measuring are really exposures to the dollar return of a yen bond with a maturity
of three months. Volatility in the yen interest rate affects the measured exposure.
To the extent that hedging takes place by borrowing at that rate, however, we are
measuring the appropriate exposure. If, alternatively, investors borrow at an
overnight rate so that they bear no interest rate risk, the appropriate measure of
the exposure is the return on the spot exchange rate overnight. This measure is
provided in the fifth and sixth columns of Table 1. It does not differ significantly
from the one obtained using the three-month borrowing rate.
In this paper, we do not try to forecast exposures using variables other than
past exposures. Hence, using exchange rate forecasts to forecast exposures is left
for further work. However, using past exposures to predict future exposures, we
find that exposures follow an ARIMA(O,l,l)
process.
So far, we have estimated exposures in a variety of ways. We found that
exposures are volatile and that there is no statistical evidence that they differ
systematically from one when using daily data. We also found that exposures are
somewhat predictable using a time-series model. In Table 2, we investigate
whether the alternative ways of measuring exposures using daily data make a
difference in the performance of minimum-variances
hedges. For each way of
measuring the exposure, we report the return of the hedged portfolio, the ratio
of the volatility of the hedged portfolio, and the volatility of the yen return of the
Nikkei. Overall, it turns out that the volatility of a hedged portfolio of the Nikkei
225 is about the same regardless of how exposures are measured.
In Table 2, the daily volatility of the Nikkei 225 is 0.62 percent in yen and 0.79
percent in dollars, using the exchange rate observed in New York. Through
hedging, an American investor can hold the Nikkei 225 bearing the same volatility as a Japanese investor. The hedged returns have the interpretation
of
excess returns. Comparing the excess return on the unhedged portfolio with the
excess return on hedged portfolios, we find that hedging would have reduced
the excess return by about 0.014 percent per day. Using a minimum-variance
hedge, an investor would have reduced the dollar volatility of his Nikkei investment by slightly more than 20 percent at the cost of reducing the return by
slightiy more than 15 percent. The cost of hedging is consistent with the existence of a positive risk premium for bearing yen risk over the sample period.
RETURNS
OF HEDGED
PORTFOLIOS
It is clear from Table 2 that using an estimated beta as opposed to assuming
that beta is one has no impact on hedging performance.
The volatility of the
107
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
RETURNS
Table 2
OF HEDGED
PORTFOLIOS
Year
1
2
3
4
5
6
7
1982
1.09
(62.54)
8.80
(41.57)
5.81
(61.25)
5.37
(44.38)
16.57
(78.27)
12.00
(95.10)
10.03
(52.86)
7.88
(38.92)
8.46
(62.05)
-2.13
(59.64)
-1.38
(35.36)
-1.02
(36.62)
8.07
(41.95)
7.64
(57.38)
10.30
(52.51)
1.41
(46.40)
-4.81
(52.55)
2.23
(48.74)
-4.75
(92.71)
4.71
(56.55)
1.75
(74.80)
11.11
(57.00)
22.32
(97.00)
20.31
(102.56)
9.23
(65.89)
0.49
(67.43)
8.14
(78.90)
-0.83
(62.63)
6.96
(41.58)
4.05
(61.19)
3.49
(44.40)
15.18
(78.22)
10.78
(95.52)
8.76
(52.90)
6.37
(38.90)
6.86
(62.14)
-2.44
(63.82)
6.65
(41.60)
3.69
(61.18)
3.86
(44.43)
15.06
(78.23)
11.18
(95.46)
9.11
(52.97)
6.46
(38.90)
6.79
(62.30)
-2.10
(65.19)
6.74
(41.60)
3.78
(61.18)
3.83
(44.41)
15.20
(78.23)
11.02
(95.47)
9.02
(52.94)
6.43
(38.90)
6.89
(62.43)
-2.08
(65.20)
6.55
(41.60)
3.79
(61.18)
3.83
(44.41)
15.20
(78.23)
11.01
(95.49)
9.01
(52.93)
6.43
(38.90)
6.90
(62.43)
1983
1984
1985
1986
1987
1988
1989
All
Note:
The returns in percent are multiplied by 100. The statistics in column 1 are in
yen; all other statistics are in dollars. The first four portfolios are (1) a long
position in the Nikkei; (2) a long position in yen; (3) a long position in the
Nikkei financed in dollars at the three-month
euro-dollar rate; (4) a long
position in the Nikkei financed in yen at the 3-month euro-yen rate. The last
three portfolios all involve a long position in yen, an amount borrowed at the
three-month euro-yen rate equal to 13,with the excess invested or borrowed at
the three-month euro-dollar rate. Portfolio (5) uses the daily returns within a
month to compute 13 and uses that estimated 13 every day for the following
month. Portfolio (6) uses a rolling regression using 30 days of data to compute
p and holds the hedge for a day. Portfolio (7) uses I3 estimated from an
ARIMA (O,l,l) model that uses all p’s estimated to construct portfolio (5). In
all cases, we use the dollar/yen exchange rate observed in New York.
hedged portfolio is the same whether the exchange rate is observed in the US or
in Tokyo. Further, this volatility is the same whether betas are estimated with
leads and lags, using a thirty-day rolling regression,
or forecasted using an
ARIMA(O,l,l)
model.
One might argue that our results are due to an estimation period that is too
short. In Table 3, we show that estimating exposures over four months leads to
108
GLOBAL FINANCE JOURNAL
EXPOSURE
Year
1982
1983
1984
1985
1986
1987
1988
1989
All
ESTIMATES
Table 3
USING LONGER
1
1.1233
(0.2241)
1.1706
0.2766
1.1948
(0.3732)
0.7903
(0.3084)
1.0671
(0.3014)
0.6533
(0.4840)
0.7020
(0.3261)
0.9086
(0.2023)
0.9494
(0.3129)
ESTIMATION
3(2), 1992
PERIODS
2
1.0959
(0.1147)
1.1686
(0.1287)
1.1689
(0.1802)
0.8863
(0.1414)
0.9873
(0.1317)
0.8812
(0.1998)
0.7046
(0.1627)
0.9600
(0.0906)
0.9766
(0.1450)
Note:
This table presents the yearly averages of estimates of the exposure of the
dollar return of the Nikkei to the rate of change of the dollar price of the yen.
The estimates for each month are obtained using (1) all returns within the
month and (2) all returns within the month and within the three previous
months. The estimates measure the exposure with respect to the dollar price
of a three-month euro-yen deposit with the New York dollar price of the yen.
The standard deviations are averages of the standard deviations of the
estimates.
substantially more precise estimates of the exposures. However, the greater
precision of exposures does not lead to a noticeable improvement in the performance of the hedged portfolio. In Table 4, we compare the performance of a
hedged portfolio with a hedge ratio of -1 to portfolios that use hedge ratios
estimated using, respectively, 4 months of daily data and 20 months of monthly
data. The volatility of the hedged portfolio is compared to the volatility of a
hedged portfolio with a hedge ratio of -1. The standard deviations of the various hedged portfolios are indistinguishable.
The conclusion from our analysis is that the yen return of the Nikkei is not
exposed to the dollar/yen exchange rate, so that the dollar return of the Nikkei
has a beta with respect to the dollar/yen exchange rate of 1. Although we want
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
Table 4
THE PERFORMANCE
OF HEDGED PORTFOLIOS
USING
ESTIMATION
PERIODS FOR EXPOSURES
Year
1
109
LONGER
2
3
0.99
1.00
1.01
1984-1985
(19)
0.99
(65)
1.00
(5)
1.00
1986-1987
(24)
1.00
(95)
1.00
(24)
1.00
1988-1989
(24)
1.01
(94)
1.00
(24)
0.99
All
(24)
1.00
(96)
1.00
(24)
1.00
1982-1983
(81)
(350)
(77)
Note:
All portfolios have a long position in yen, an amount borrowed at the threemonth euro-yen rate equal to S, with the excess invested or borrowed at the
three-month euro-dollar rate. Portfolio (1) uses four months of daily data to
compute S. Portfolio (2) uses a rolling regression of 20 weeks of data. Portfolio
(3) uses monthly data for 20 months. For portfolios (1) and (3), we assume that
the hedge is maintained for a month, whereas for (2) we assume it to be held
for a week. All returns assume that the hedge is maintained for a month. In all
cases, we use the dollar/yen exchange rate observed in New York. We report
the standard deviation of the portfolio divided by the standard deviation of a
portfolio that uses a hedge ratio of 1 and is held for the same period of time.
We report this ratio for two-year averages; the number of observations differs
across cells and is reported in parentheses.
to be cautious in generalizing from this example, it suggests that little is to be
gained by investigating the own currency exposure of market indices. After all,
the gain from exploiting own currency exposures of market indices other than
the Nikkei is likely to be extremely small if such exposures are nontrivial because
these indices are unlikely to be a large component of internationally diversified
portfolios.
Optimal
hedge ratios.
Given our previous evidence, we can restate the optimal investment in yen in
excess of the amount borrowed in yen to finance the investment in the Nikkei as
WY
=
(a - R,)(PV-~I~P&
(5”)
110
GLOBAL FINANCE JOURNAL
3(2), 1992
since p’ (in this case, the exposure of an investment in the Nikkei 225 financed in
yen) is effectively zero for the Nikkei 225. wy is the investor’s optimal yen
holdings in excess of the yen position required to finance his investment in the
Nikkei if the investor can invest in the dollar risk-free asset, the yen risk-free
asset, and the Nikkei. Hence, if it is optimal for the investor to invest half his
wealth in the Nikkei, the optimal yen holdings would be wy - 0.5 since the
investor would have to borrow the equivalent of half his wealth in Japan to
finance his investment in the Nikkei. The resulting optimal hedge ratio is (wy wN)IwN, where wN is the fraction of wealth invested in yen to finance the zero
net investment portfolio long in the Nikkei. If pf = 0, the investor invests only in
a zero net foreign investment portfolio long in the Nikkei; and the optimal hedge
ratio is - 1. In Table 5, we report optimal hedges constructed using equation (5”)
assuming that, for each year, the expected returns and the variances of long
positions in a zero net foreign investment portfolio long in the Nikkei and in a
euro-yen deposit are equal to their ex post value. We choose as the investor’s
target return the expected return on an unhedged position in the Nikkei, assuming that the expected rate of change of the dollar/yen exchange rate and its
COMPOSITION
Estimation
1982
1983
1984
1985
1986
1987
1988
1989
1982-1989
Period
Table 5
AND PERFORMANCE
OF MEAN-VARIANCE
EFFICIENT PORTFOLIO
Nikkei weight
Yen weight
Hedge ratio
Var. ratio
0.38
0.53
0.12
0.60
0.77
0.54
0.99
0.00
0.94
0.75
-0.80
-0.67
0.66
0.71
0.97
0.10
-0.08
0.39
0.99
-2.49
-6.48
0.10
-0.09
0.79
-0.90
-20.76
-0.59
0.38
0.53
0.12
0.60
0.77
0.54
0.99
0.01
0.93
Note:
Optimal portfolio and performance relative to portfolio fully invested in Nikkei with hedge ratio of - 1. The position in the Nikkei is fully hedged against
exchange rate risk using-a hedge ratio of - 1. The position in yen is a position
in a three-month euro-yen deposit and ignores the amount borrowed to hold
the Nikkei. The hedge ratio is defined as (euro-yen deposit weight-Nikkei
weight)/(Nikkei
weight). The var. ratio is the ratio of the variance of the
optimally hedged portfolio and of the unhedged portfolio. The hedged portfolio is constructed assuming that the investor uses as the ex ante forecasts of
means, variances, and covariances their ex post values during the estimation
period.
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
111
volatility equal their ex post values. Hence, the table provides a mean-variance
efficient portfolio invested in a hedged position in the Nikkei and a currency
position that yields the same expected return as an unhedged position in the
Nikkei.
It is clear from Table 5 that the optimal hedge changes substantially over time.
However, the hedge ratios used assume for some years that the expected return
on a hedged position in the Nikkei will be negative, since ex post the return was
sometimes negative. One interpretation of the variance ratios is that they show
the decrease in variance that could have been achieved through investment in
yen assuming that otherwise the investor would have held a position in the
Nikkei unhedged. One might choose a longer period of time to measure the
means and variances to form expectations about the future than one year to gain
another perspective on the benefits of hedging. If one uses the whole sample
period to form estimates of means and variances, the optimal hedge ratio would
be -0.59, which is substantially smaller than one in absolute value. However,
the variance reduction brought about by selecting that hedge ratio instead of a
hedge ratio of -1 is remarkably small-about
7 percent. On average, investments in a euro-yen deposit would have had a small excess return relative to
investing in a euro-dollar deposit. This implies that reducing the hedged investment in the Nikkei and increasing the investment in the euro-yen deposit
by equivalent amounts reduces the portfolio’s expected return. To keep the
expected return of the portfolio constant, one has to increase investment in the
euro-yen deposit substantially more than one has to decrease the hedged investment in the Nikkei. Consequently,
even a substantial departure of the
hedge ratio from -1 has only a moderate impact on the performance of the
portfolio.
Rather than using the ex post cost of hedging to construct hedge ratios, we
could use a measure of the ex ante cost derived from a model for the pricing of
forward contracts. For instance, suppose that the appropriate world market
portfolio is simply the world market portfolio of equities with returns measured
in dollars. In this case, if the Nikkei is 40 percent of the world market portfolio,
our evidence suggests that the beta of the yen would be 0.4. Hence, if the market
portfolio earns a risk premium of lo%, the ex ante cost of hedging is 4%. Since
such a hedging cost turns out to be not very different from the whole sample ex
post hedging cost, using this ex ante measure yields a similar hedge ratio as the
whole sample hedge ratio obtained in Table 5.
Whereas ex post the yen forward rate is on average lower than the spot rate at
maturity of the forward contract, it is also clear that the forward price is sometimes higher than the realized spot rate for a substantial period of time. Hence,
the extent to which one can construct a portfolio with currency positions that
dominate a portfolio of investments
in hedged equity investments
depends
crucially on how well one can forecast exchange rates relative to the forward
rate. If one can forecast that the spot exchange rate will differ from the forward
rate with some degree of precision, then a hedge ratio of -1 is not optimal.
Recent empirical literature on the risk premium embedded in forward prices
suggests that this risk premium can be forecasted.13
112
GLOBAL FINANCE JOURNAL
3(2), 1992
CONCLUSION
This paper provides a simple formula for mean-variance efficient hedging and
applies it to an investment in the Nikkei. We find that there is no gain from
trying to get the best possible estimate of the Nikkei’s exposure to the dollar/yen
exchange rate. In particular, the standard deviation of tl.? return of hedged
portfolios is generally lowest when the exposure is assumed to be one rather
than when the exposure is estimated from historical data. If some gain can be
achieved through an optimal choice of currency positions, it is through better
estimates of the cost of hedging. The expected return cost of hedging is the
difference between the expected dollar return from holding a euro-yen deposit
and the dollar return from holding a euro-dollar deposit. If this expected return
difference is positive, there is a positive reward for bearing exchange rate risk
and hedging is costly.
NOTES
1. See, for instance, Arnott and Henrikkson [3], Black [5], Celebuski, Hill and
Kilgannon [7], Jorion [ll], Perold and Shulman L [14] and Kaplanis and
Schaefer [ 121.
2. For instance, Perold and Schulman [14] argue that exposures are equal to
one whereas Adler and Simon [l] and Kaplanis and Schaefer [12] find exposures significantly greater than one using weekly data for a subsample at
the beginning of the nineteen eighties.
3. See Stulz [16] for more details.
4. See Stulz [16] and Kaplanis and Schaefer [12].
5. This result applies only to assets that are nominally risky, however.One
cannot construct zero net foreign investment portfolios in the nominally
riskless asset, i.e., the return on a long investment in the riskless asset of
country i financed by a short investment in that asset is identically equal to
zero.
6. This is the model used by Black [5]. For a detailed analysis of the conditions
under which universal hedging ratios obtain, see Adler and Prasad [2].
7. The formula we obtain holds for any numeraire with some obvious adjustments.
8. See Stulz [16].
9. See Adler and Simon [l] and Kaplanis and Schaefer [12] for evidence on the
exposure of own currency returns to non-dollar exchange rates to using
weekly data.
10. In addition to the difference in mean betas, 17 betas using exchange rates
observed in Tokyo are significantly greater than one, whereas only three
betas using exchange rates observed in New York are significantly greater
than one.
11. Bailey and Stulz [4] show that the lack of synchronization
between US and
Japanese index returns biases the correlation between these returns downward.
Optimal Hedging of Stock Portfolios Against Foreign Exchange Risk
12. See Dimson [B] and Fowler and Rorke [9].
13. See, for instance, Cumby [7], Engel and Hamilton
113
[lo] and Mark [13].
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